Inerter ($F=b({\ddot{x}}_1-{\ddot{x}}_2)$)

An inerter is an ideal two-terminal mechanical element whose equal and opposite terminal forces are proportional to the relative acceleration of its terminals. With displacements $x_1,x_2$, define the forces applied to the device asThe parameter $b$ is its mechanical inertance and has units of mass. Both terminals can move; if their accelerations are identical, this ideal constitutive law gives zero terminal force. Malcolm C. Smith introduced the element in a 2002 mechanical-network synthesis paper.^[ref]

The power and stored-energy relation explains its role. Let $v={\dot{x}}_1-{\dot{x}}_2$. ThenAn ideal inerter stores and returns energy; this relation is not viscous dissipation. For an original force calculation, $b=100\mathrm{kg}$ and relative acceleration $0.2\mathrm{m}{\mathrm{s}}^{-2}$ give a 20-newton terminal-force magnitude.

A rotational realization also shows why inertance need not equal the assembly's translational mass. In an idealized rack and pinion driving a flywheel, suppose relative terminal speed $v$ produces angular speed $\omega =v/r$. A flywheel with moment of inertia $J$ has energyso its ideal equivalent inertance is $b=J/r^2$. Choosing $J=0.02\mathrm{kg}{\mathrm{m}}^2$ and $r=0.01\mathrm{m}$ gives $b=200\mathrm{kg}$. This is a transmission-based example, not an assertion that the physical flywheel has a 200-kilogram mass.^[ref]

Friction, backlash, compliance and finite travel alter real-device behavior. An inerter can be combined with a spring or viscous damper, but their constitutive roles remain different: relative acceleration, displacement and velocity enter different element laws.

Mechanical inertance $b$ is the coefficient relating the terminal force of an ideal inerter to relative terminal acceleration:The forces here are applied to the device. Its mechanical inertance has SI units $\mathrm{N}{\mathrm{s}}^2{\mathrm{m}}^{-1}=\mathrm{kg}$. The constitutive law was introduced by Malcolm C. Smith as a two-terminal mechanical-network element.^[ref]

Mechanical inertance is a response coefficient and need not equal the assembly's translating mass. For an ideal rack and pinion driving a flywheel, let relative terminal speed $v$ produce angular velocity $\omega =v/r$. Comparing stored energies giveswhere $J$ is the flywheel's moment of inertia and $r$ is the pinion's pitch radius. In an original example, $J=0.0125\mathrm{kg}{\mathrm{m}}^2$ and $r=0.005\mathrm{m}$ give $b=500\mathrm{kg}$. A relative acceleration of $0.30\mathrm{m}{\mathrm{s}}^{-2}$ then requires150newtons. Halving $r$ with $J$ fixed multiplies $b$ by four; it does not multiply the physical mass by four.

At relative speed $0.20\mathrm{m}{\mathrm{s}}^{-1}$, the same ideal element stores10joules. If both terminals instead have identical acceleration, the ideal inertial force is zero regardless of their common acceleration. This two-terminal behavior distinguishes the coefficient from simply attaching a mass to a fixed reference.

A measurement can estimate mechanical inertance from the slope of force against relative acceleration over a range where the ideal model is adequate. Friction, backlash and mechanical compliance can produce additional force components, so one force-to-acceleration ratio at an arbitrary operating point need not identify a constant inertance.